$7.$ This question is based on Mookherjee and Png $(1995).$ You will find it useful to review the class notes on that paper as you work through this question. $(10$ points$)$

$-$ Consider the model of Mookerjee and Png$,$ with the cost functions $\backslash ($ c$($w$)=2\backslash$sqrt$\{$w$\}\backslash )$ and $\backslash ($ e$(\backslash$mu$)=\backslash$frac$\{\backslash$mu$^\{2\}\}\{2\}\backslash ).$

$-$ Assume that there are no penalties for bribery $($i$.$e$.\backslash (\backslash$lambda$=0\backslash )).$

$-$ The harm from waste: $\backslash ($ h$($w$)=$h w $\backslash ),$ where $\backslash ($ h$>0\backslash )$ is the marginal cost of harm to the environment per unit of waste.

Suppose that the regulator wants to try a policy of "moral persuasion;" that is$,$ conduct a campaign to encourage people to the right thing voluntarily. Under this policy, the regulator encourages firms to consider $($become aware$)$ the environmental damage of their production. Assuming that the policy is effective the firm incurs a cost of $\backslash (\backslash$alpha h w $\backslash ),$ where $\backslash (\backslash$alpha $\backslash$in$(0,1)\backslash ).$ Note that if $\backslash (\backslash$alpha$=1\backslash )$ then the firm would choose a waste level that is optimal for society so that the moral policy works perfectly. If $\backslash (\backslash$alpha$=0\backslash )$ the firm does not account for the harm and the policy is useless. So assume that $\backslash (0<mi\; id="EditorElement\_3934956">\backslash </mi>$alpha

Using backward induction, solve for the sub$-$game perfect Nash equilibrium of the game under this policy of moral persuation. Specifically,

$($a$)$ Derive the equilibrium level of bribe $\backslash ($ b $\backslash )$ under this policy. Is the bribe larger or smaller under this policy $($relative to the case without this policy of moral persuation$)?$

$($b$)$ Use your answer to $($a$)$ to derive the best responses of the inspector and the firm under the policy.

$($c$)$ Determine whether the equilibrium waste is higher or lower under this moral persuation policy $($relative to the case without it$).$ You may use graphs, similar to those we utilized in class, to answer this question.

$($d$)$ Based on your answer to $($c$)$ comment on the effectiveness of this policy both in terms of whether it reduces waste and in terms of bribery. HINT: If you cannot complete all the algebra for this question $-$ don't worry. Think about how bribery worked in our analysis in class, and whether it had an adverse effect on waste. Use that logic to answer this question.